This study will last at least 6 months. Participants will first be screened to verify diagnosis of aplastic anemia. The screening will include a physical examination, blood test, bone marrow biopsy from the pelvic bone, and review of medications and medical history. Individuals who are eligible will then start the first treatment period. Participants will receive two medications: cyclosporine will be taken twice a day and sirolimus will be taken once a day. Depending on side effects, the doses of either drug may be temporarily stopped or lowered. On Day 1, blood will be drawn and females will undergo a pregnancy test. Subsequent study visits will occur weekly for the first month, every 2 weeks for 2 months, and then once a month for the remainder of the study. Each visit will include a physical examination, vital sign assessment, and review of side effects and medications. Blood tests will be performed weekly for the first 3 weeks, and then every 2 weeks.